Operator norm convergence of spectral clustering on level sets

Journal of Machine Learning Research

Volumn 12, Issue , 2011, Pages 385-416

  Pelletier, Bruno ^a   Pudlo, Pierre ^b  

^a UNIVERSITÉ RENNES 2   (France)

^b UNIVERSITÉ DE MONTPELLIER   (France)

Author keywords

Connected components; Graph; Level sets; Spectral clustering; Unsupervised classification

Indexed keywords

CONNECTED COMPONENT; GRAPH; LEVEL SET; SPECTRAL CLUSTERING; UNSUPERVISED CLASSIFICATION;

CONVERGENCE OF NUMERICAL METHODS; LAPLACE EQUATION; LAPLACE TRANSFORMS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA;

CLUSTERING ALGORITHMS;

EID: 79952743200     PISSN: 15324435     EISSN: 15337928     Source Type: Journal    
DOI: None     Document Type: Article

References (38)

1. M. R. Anderberg. Cluster Analysis for Applications. Academic Press, New-York, 1973.
2. A. Azzalini and N. Torelli. Clustering via nonparametric estimation. Stat. Comput, 17:71-80, 2007.
3. M. Belkin and P. Niyogi. Towards a theoretical foundation for Laplacian-based manifold methods. In Learning Theory, volume 3559 of Lecture Notes in Comput. Sci., pages 486-500. Springer, Berlin, 2005.
4. M. Belkin, I. Matveeva, and P. Niyogi. Regularization and semi-supervised learning on large graphs. In Learning Theory, volume 3120 of Lecture Notes in Comput. Sci., pages 624-638. Springer, Berlin, 2004. (Pubitemid 38940364)
5. G. Biau, B. Cadre, and B. Pelletier. A graph-based estimator of the number of clusters. ESAIM Probab. Stat, 11:272-280, 2007. (Pubitemid 47177024)
6. F. R. K. Chung. Spectral Graph Theory, volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.
7. R. R. Coifman and S. Lafon. Diffusion maps. Appl. Comput. Harmon. Anal, 21:5-30, 2006. (Pubitemid 43947619)
8. A. Cuevas, M. Febrero, and R. Fraiman. Estimating the number of clusters. Canadian Journal of Statistics, 28:367-382, 2000.
9. A. Cuevas, M. Febrero, and R. Fraiman. Cluster analysis: a further approach based on density estimation. Comput. Statist. Data Anal., 36:441-459, 2001. (Pubitemid 32589428)
10. R. O. Duda, P. E. Hart, and D. G. Stork. Pattern Classification. Wiley Interscience, New-York, 2000.
11. M. Fillipone, F. Camastra, F. Masulli, and S. Rovetta. A survey of kernel and spectral methods for clustering. Pattern Recognition, 41(1):176-190, 2008. (Pubitemid 47284212)
12. L. A. Garćia-Escudero, A. Gordaliza, C. Matrán, and A. Mayo-Iscar. A general trimming approach to robust cluster analysis. Ann. Statis., 36(3):1324-1345, 2008.
13. E. Giné and V. Koltchinskii. Empirical graph Laplacian approximation of Laplace-Beltrami operators: large sample results. In High Dimensional Probability, volume 51 of IMS Lecture Notes Monogr. Ser., pages 238-259. Inst. Math. Statist., Beachwood, OH, 2006.
14. J. A. Hartigan. Clustering Algorithms. Wiley, New-York, 1975.
15. T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New-York, 2001.
16. M. Hein, J.-Y. Audibert, and U. von Luxburg. Graph Laplacians and their convergence on random neighborhood graphs. Journal of Machine Learning Research, 8:1325-1368, 2007.
17. J. Jost. Riemannian Geometry and Geometric Analysis. Universitext. Springer-Verlag, Berlin, 1995.
18. T. Kato. Perturbation Theory for Linear Operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
19. A. N. Kolmogorov and V. M. Tikhomirov. ε-entropy and ε-capacity of sets in functional space. Amer. Math. Soc. Transl. (2), 17:277-364, 1961.
20. V. I. Koltchinskii. Asymptotics of spectral projections of some random matrices approximating integral operators. In High Dimensional Probability (Oberwolfach, 1996), volume 43 of Progr. Probab., pages 191-227. Birkhäuser, Basel, 1998.
21. T. Linder. Learning-theoretic methods in vector quantization. In Principles of Nonparametric Learning (Udine, 2001), volume 434 of CISM Courses and Lectures, pages 163-210. Springer, Vienna, 2002.
22. J. MacQueen. Some methods for classification and analysis of multivariate observations. In Proc. Fifth Berkely Symp. Math. Statist. Prob., volume 1, pages 281-297, 1967.
23. G. McLachlan and D. Peel. Finite Mixture Models. Wiley, New-York, 2000.
24. S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag, London, 1993.
25. J. W. Milnor. Morse Theory. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N. J., 1963.
26. B. Nadler, S. Lafon, R. R. Coifman, and I. G. Kevrekidis. Difusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmon. Anal., 21(1):113-127, 2006. (Pubitemid 43947618)
27. A. Ng, M. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In T. Dietterich, S. Becker, and Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14, pages 849-856. MIT Press, 2002.
28. D. Pollard. Consistency of k-means clustering. Ann. Statis., 9(1):135-140, 1981.
29. D. Pollard. Convergence of Stochastic Processes. Springer-Verlag. New-York, 1984.
30. B. L. S. Prakasa Rao. Nonparametric Functional Estimation. Probability and Mathematical Statistics. Academic Press Inc., New York, 1983.
31. L. Rosasco, M. Belkin, and E. de Vito. On learning with integral operators. Journal of Machine Learning Research, 11:905-934, 2010.
32. A. W. van der Vaart. Bracketing smooth functions. Stochastic Process. Appl., 52(1):93-105, 1994.
33. A. W. van der Vaart. Asymptotic Statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998.
34. A. W. van der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes. Springer Series in Statistics, Second Edition. Springer, New-York, 2000.
35. U. von Luxburg. A tutorial on spectral clustering. Stat. Comput., 17(4):395-416, 2007.
36. U. von Luxburg and S. Ben-David. Towards a statistical theory of clustering. In PASCAL Workshop on Statistics and Optimization of Clustering, 2005.
37. U. von Luxburg, M. Belkin, and O. Bousquet. Consistency of spectral clustering. Ann. Statis., 36(2):555-586, 2008.
38. L. Zelnik-Manor and P. Perona. Self-tuning spectral clustering. In Eighteenth Annual Conference on Neural Information Processing Systems (NIPS), 2004.